FOM: a question re. completeness

[michael Detlefsen]

A QUESTION CONCERNING COMPLETENESS

I'd be interested to hear what others on FOM might have to say about the
following questions. I've been thinking about them as I write my book on
Goedel's theorems. Please forgive me if there are already answers to them
that I should know of.

Preamble: The sentences unprovable in PA all seem to have the following
property: if they were to be provable in PA, PA would be inconsistent.
Sentences unprovable in other well-known systems (e.g. ZF) do not all
appear to have this property.

This suggests a kind of completeness that is related (though not, of
course, equivalent) to Post completeness. We might (without much
inspiration) call this kind of completeness 'consistency-completeness' or
'weak Post completeness' or some such thing and define it as follows: T is
consistency-complete iff for every sentence s of the language of T that is
not provable in T, if s were provable in T, T would be inconsistent.

Questions:

(1) Is PA consistency-complete? If so, is there a proof of this? If not,
what is the counterexample?

(2) Is there a proof that ZF is not consistency-complete?

(3) Is consistency-completeness an important or interesting property? Is
there virtue (or something else of interest) in a theory's being
consistency-complete? If so, what is the best way to describe this virtue?




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Michael Detlefsen
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana  46556
U.S.A.
e-mail:  Detlefsen.1 at nd.edu
FAX:  219-631-8609
Office phone: 219-631-7534
Home phone: 219-232-7273
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